| L(s) = 1 | − 2-s − 4-s + 3·8-s + 2·11-s − 16-s + 6·19-s − 2·22-s − 6·23-s + 6·29-s + 6·31-s − 5·32-s + 6·37-s − 6·38-s + 8·41-s − 6·43-s − 2·44-s + 6·46-s + 8·47-s − 7·49-s − 12·53-s − 6·58-s − 2·59-s + 6·61-s − 6·62-s + 7·64-s − 12·67-s + 2·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 0.603·11-s − 1/4·16-s + 1.37·19-s − 0.426·22-s − 1.25·23-s + 1.11·29-s + 1.07·31-s − 0.883·32-s + 0.986·37-s − 0.973·38-s + 1.24·41-s − 0.914·43-s − 0.301·44-s + 0.884·46-s + 1.16·47-s − 49-s − 1.64·53-s − 0.787·58-s − 0.260·59-s + 0.768·61-s − 0.762·62-s + 7/8·64-s − 1.46·67-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.512324778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.512324778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.66729373234460, −14.22646877159252, −13.92704726676283, −13.35104915263439, −12.81615760638356, −11.98858438968003, −11.85919536413442, −11.06050857283707, −10.49478213739487, −9.893021485793235, −9.558130618106436, −9.105623227222873, −8.389938556787715, −7.904527672144321, −7.574266570271454, −6.721641812159882, −6.180549394713673, −5.537657431250829, −4.682124440223937, −4.417374238497121, −3.588161907785631, −2.930582209070850, −1.988583998478360, −1.178707587635554, −0.5996235459618626,
0.5996235459618626, 1.178707587635554, 1.988583998478360, 2.930582209070850, 3.588161907785631, 4.417374238497121, 4.682124440223937, 5.537657431250829, 6.180549394713673, 6.721641812159882, 7.574266570271454, 7.904527672144321, 8.389938556787715, 9.105623227222873, 9.558130618106436, 9.893021485793235, 10.49478213739487, 11.06050857283707, 11.85919536413442, 11.98858438968003, 12.81615760638356, 13.35104915263439, 13.92704726676283, 14.22646877159252, 14.66729373234460
